It's been a while since I've read Don Juan, but from what I can remember -
Byron isn't talking about Julia apologizing here.
This stanza happens in between Alfonso first showing up in Julia's bedroom under the impression that Juan is there, which he is. Alfonso finds nothing and leaves. The stanza in question happens after Antonia suggests putting Juan in the closet before Alfonso returns.
'A lady with apologies abounds;—
It might be that her silence sprang alone
From delicacy to Don Juan's ear,
To whom she knew his mother's fame was dear.'
From what I can remember, this stanza specifically is Byron making reference to the fact that previously, Julia had complained about Alfonso & Inez and also lied/made up excuses about herself and Juan to ease Alfonso's suspicions, but is strangely silent there and then. The stanza's end pretty much says
"Julia usually had infinite excuses and apologies, but her silence at the moment might be because Don Juan loves and respects his mother and Julia doesn't want him to hear her slander Inez (in regard to his mother's love of Alfonso), so she stays silent despite it ruining her alibi."
In short: 'a lady with apologies abound' does mean 'a lady full of apologies', but not in the sense that she is apologizing at that point in time in the story. Instead, it's more of Alfonso's observation that a core personality trait of Julia's is strangely absent, despite her previous excuses/lies.
