It's probably a Biblical reference.
Many western poems and works of literature make allusion or references to the Christian/Jewish Bible, or borrow its symbolic language.
In the Bible, Western Winds are used as a symbol of God's blessing, refreshing, provision, and healing. Eastern Winds are symbolic of trouble, famine, exhaustion, difficulties, and even judgement.
This is because the Bible usually speaks from a Israel-centric viewpoint, and Israel is located between the Mediterranean Sea to the West and the deserts of the Arabian peninsula to the East. Eastern Winds brought dry desert heat that killed crops, brought famine, and made summers miserable, Western Winds brought refreshing rains and cool pleasant air from the sea and the promise of good harvest and plenty of food to eat.
"Ephraim feeds on wind,
And pursues the east wind continually;
He multiplies lies and violence." - Hosea 12:1
"Though he flourishes among the reeds,
An east wind will come,
The wind of the Lord coming up from the wilderness [read: desert];
And his fountain will become dry
And his spring will dry up;
It will plunder his treasury of every precious article." - Hosea 13:15
"And when the sun came up God designated a scorching east wind, and the sun beat down on Jonah’s head so that he became faint, and he begged with all his soul to die, saying, “Death is better to me than life!”" - Jonah 4:8
"Then behold, seven ears, thin and scorched by the east wind, sprouted up after them." - Genesis 41:6
"Should a wise man answer with windy knowledge,
And fill himself with the east wind?
Should he argue with useless talk,
Or with words which do not benefit?" - Job 15:2
"Terrors overtake him like a flood;
A storm steals him away in the night.
The east wind carries him away, and he is gone;
For it sweeps him away from his place.
For it will hurl at him without mercy;
He will certainly try to flee from its power." - Job 27:21
"Panic seized them there,
Anguish, as that of a woman in childbirth.
With the east wind
You smash the ships of Tarshish." - Psalm 48:7
(This looks like provision, but actually in-context is a judgement, both in-context in the Psalm, and in the context of when it actually occurred in Exodus. Link gives expanded context)
"He made the east wind blow in the sky
And by His power He directed the south wind.
When He rained meat upon them like the dust,
Even winged fowl like the sand of the seas,
He let them fall in the midst of their camp,
All around their dwellings.
So they ate and were well filled,
And He satisfied their longing." - Psalm 78:26
"You contended with them [b]by banishing them, by driving them away.
With His fierce wind He has expelled them on the day of the east wind." - Isaiah 27:8
"To make their land a desolation,
An object of perpetual hissing;
Everyone who passes by it will be astonished
And shake his head.
Like an east wind I will scatter them
Before the enemy;
I will show them My back and not My face
In the day of their disaster.’”" - Jeremiah 18:18
(West wind providing deliverance:)
"So the Lord shifted the wind to a very strong west wind, which picked up the locusts and drove them into the Red Sea; not one locust was left in all the territory of Egypt." - Exodus 10:19
(West wind (from the sea) bringing wind to end extended famine:)
"Now Elijah said to Ahab, “Go up, eat and drink; for there is the sound of the roar of a heavy shower.” [A prophetic declaration - no rain has occurred in 3 years, but Elijah is saying he prophetically "hears" what has not yet arrived] So Ahab went up to eat and drink. But Elijah went up to the top of Carmel; and he bent down to the earth and put his face between his knees. 43 And he said to his servant, “Go up now, look toward the sea.” So he went up and looked, but he said, “There is nothing.” Yet Elijah said, “Go back” seven times. And when he returned the seventh time, he said, “Behold, a cloud as small as a person’s hand is coming up from the sea.” And Elijah said, “Go up, say to Ahab, ‘Harness your chariot horses and go down, so that the heavy shower does not stop you.’” Meanwhile the sky became dark with clouds and wind came up, and there was a heavy shower [breaking the multi-year famine]." - 1 Kings 18:41-46
God did use the desert wind (east wind) to provide deliverance in parting the red sea for Moses in the book of Exodus, but in that case, the dry east desert wind was what they needed to dry up the water. (Exodus 14:21) Still is kinda the exception to west wind = good, east wind = bad, but it's the exception that proves the rule.
These verses, and others like them, have provided Christianity with symbolic language of West vs East winds that have been used in the 2000 years since.
Here's a modern Christian song using the same symbolism:
Awake awake o' North wind, Awake awake o' South wind, Blow over me.
Come o' Winds of Testing, Come o' Winds of Refreshing, Blow over me.
I won't be afraid, I will face the Wind. Let the winds blow! Let the winds blow!
(The Winds of Testing = Eastern Arabian desert winds, Winds of Refreshing = Western Mediterranean sea winds)
I can't speak for certain that this is a Biblical reference, East and West winds are a very well-known example of Bible symbolism, so much so I originally thought your Question was asked on Christianity.SE or BiblicalHermeneutics.SE when I read the title.
I would read your poem as, "Times of refreshing, when wilt thou arrive?", he's asking for God's providence, or Fate/Destiny to smile on him, a "changing of the winds" metaphorically. When will the dry winds stop blowing on my life, and when will times of goodness return? If only my love were in my arms! And a pleasant bed to rest on! If only the winds of fate would shift in favorable ways!
You might be more familiar with the use of seasons instead of winds to convey the same metaphores. e.g. "Spring blossoms, when wilt thou bloom? The winter of my life's hardship drags ever on!"