"I had to go see about a girl"
Since "see about" can be used as a verb phrase, the grammar is in fact correct. I can however see why you feel this phrase is unnatural - it's certainly not commonly used.
The following definition (in particular part b) should give you the right understanding.
see about,
a. to investigate; inquire about.
b. to turn one's attention to; take care of: He said he would see about getting the license plates.
In other words, quote could be rewritten more formally as:
I must turn my attention to a [certain] girl.