My girlfriend is getting her trust fund distributed of17K$Each year till she is 21 from age 18.That being said-could she gift each years worth to avoid taxes on the trust

No.  Gift tax is paid by the giver (when applicable)  and it doesn't change the requirement to include income on her personal return.

Only income earned by and distributed from the trust is taxable.  For more information she should seek legal council for all the details about the trust and how it affects her.