She would have recognized the name Guinan at the time OP is asking about (22nd century), but it may or may not have been her birth name.
She was openly going by "Guinan" as early as the 19th century in "Time's Arrow":
DOORMAN: Good afternoon, sir.
DATA: Good afternoon. I would like to speak to Guinan.
DOORMAN: And you are?
DOORMAN: Mister Data. Could it be under another name?
DOORMAN: I can't seem to find your name on the guest list, sir.
DATA: I am a personal friend.
DOORMAN: Madam Guinan has discovered many personal friends since the newspaper announcement, but if your name is not on the guest list, there's nothing I can do.
GUINAN: What exactly are you?
DATA: Android. Artificial life form.
GUINAN: Ah. Did my father send you here? Because if he did, you must go back and tell him I've not done listening
DATA: I was not sent by your father. Circumstances demand that I take you into my confidence. I require your assistance.
(Guinan touches his nose)
DATA: I am from the twenty fourth century, where you and I serve aboard a starship.
DATA: Our ship encountered a species who appears to be threatening 19th century Earth. I investigated, and was inadvertently pulled into their temporal vortex. When I saw your photograph in the newspaper, I assumed you had joined me from the future, from the Enterprise. I knew your species was long-lived, but I did not realise you had visited Earth so long ago.
Star Trek: The Next Generation 5x26 "Time's Arrow" (part 1)
Based on the line about her father, I suspect that Q was correct in stating that "Guinan" was not her real name, because I find it hard to believe that she would hide from her father under her real name. However, this is entirely speculative; I can't find any solid evidence that she ever went by any other name. Given the remoteness of Earth at this time, it is entirely possible she did not expect her father to find her there regardless of what she called herself.
It is also possible that her real name has always been "Guinan," but at some point she felt the need to lie about it to Q.
This is the same Guinan, not an alternate timeline:
GUINAN: You just do. Do you remember the first time we met?
PICARD: Of course.
GUINAN: Don't be so sure. I just mean, if you don't go on this mission, (the [drink she was making] evaporates) we'll never meet.