Working as a lecturer at a public university is considered working for the goverment?

What is the context of your question? Did you receive a W-2 or other reporting document from the government?

There is a G on box 12.

Which form did you receive; W-2 or 1099-MISC?

Possibly expanding on your facts would be helpful:

• Are you a visiting lecturer
• Do you have a contract with the university?
• Are you full-time, part-time?

Not visiting - Full time lecturer. 

All Code G means is that you contribute to a Section 457(b) plan. It has nothing to do with the government.

G — Elective deferrals and employer contributions (including non-elective deferrals) to a Section 457(b) deferred compensation plan.

A public university is not a government agency, so no, that would not be considered working for the government.