Well… if a guy makes the first move and is pursuing her on GS honestly… and the girl is single… I think that girl would at least be curious if the guy is seriously interested in her. Now if she will pursue you back will depend on her thoughts on an online friendship/relationship with the opposite gender.
I don’t think she’d consider it as regular everyday GS business, especially if you ask questions about her life. That’s different from the usual messing around or jokes.