It's a double pun on immaculate conception and general. The Roman Catholic dogma of immaculate conception says that Mary is free from original sin right from the moment of her conception.
The narrator is saying that during his father's wartime absences, he would come up with an idealized image of him. This is a conception in the sense of "a complex product of abstract or reflective thinking" (sense 2c on the Merriam-Webster page). It was immaculate, i.e., spotless, because the narrator would attribute all good qualities to the picture he built up of his father in his mind.
This idealized picture would be shattered by his father's actual behavior upon his returning home for leave. His father is a brigadier, which in the British army is the lowest-ranking of the generals. Directly after the passage you quote, the narrator goes on to say:
Like all men not really up to their job, he was a stickler for externals and petty quotidian things; and in lieu of an intellect he had accumulated an armoury of capitalized key-words like Discipline and Tradition and Responsibility. If I ever dared—I seldom did—to argue with him, he would produce one of those totem words and cosh me with it, as no doubt in similar circumstances he quelled his subalterns. If one still refused to lie down and die, he lost, or loosed, his temper. His temper was like a red dog, and he always had it close to hand.
So within a couple of days after his father's return, the narrator would see him as he really was: an incompetent, unintelligent, bad-tempered man. His father would behave toward him in the same way that he, in his capacity as general, would behave toward his subordinates in the army. So he would generally shatter the immaculate conception that the narrator has built up. In this context, "generally" means "wholly" or "entirely" (from definition 1 of entry 1 on the Merriam-Webster page), but it also means in the manner of a general, referring to his father's rank.
As Tsundoku mentions in the comments, the immaculate conception is not the same as the virgin birth. Jesus's being born of a virgin (because Mary bore the child of God without sexual intercourse) is a different Catholic doctrine from the immaculate conception.
Edit 20230202. WS2's comment caused me to revisit this answer and I realized that I had missed what is in hindsight blindingly obvious. Etymologically, generally comes from genus/genera, and is thereby related to procreation and conception:
general (adj.) c. 1200, "of wide application, generic, affecting or involving all" (as opposed to special or specific), from Old French general (12c.) and directly from Latin generalis "relating to all, of a whole class, generic" (contrasted with specialis), from genus (genitive generis) "stock, kind" (from PIE root *gene- "give birth, beget," with derivatives referring to procreation and familial and tribal groups).
The father's being a brigadier-general ties into this rather elaborate pun. And of course he has begotten the narrator, who is of a different generation ... but perhaps seeing that as part of the pun is stretching things a bit.