Yes, if it wanted to, but it would need the cooperation of President Erdogan. The steps that need to occur are as follows:
- President Erdogan to ratify the protocol and submit it to the Speaker of the Grand National Assembly.
- The Speaker to forward the protocol to the relevant parliamentary committee(s) for review.
- The committee(s) to refer the protocol back to the Assembly for approval.
- The Assembly to vote on a bill approving the ratification of the protocol.
- President Erdogan to grant Presidential Assent to the said bill.
Most of these steps could be completed fairly quickly. It might be helpful to consider the process followed with respect to the entry of North Macedonia into NATO in 2019. President Erdogan transmitted the protocol to the Speaker on May 9th, the bill approving the ratification was voted on in the Assembly on July 11th, and presidential assent was granted on July 24th; a total of 76 days.
In the present case, this would take us to the end of April. The 2023 election date has not been set in stone yet, but Erdogan has announced that he intends to hold the elections on May 14th. In 2018 the Assembly dissolved 40 days before election day.
However, in the case of North Macedonia, Erdogan, the Speaker, and the various parliamentary committees came under criticism - from the Nordic Research and Monitoring Network, for example - accused of stalling the referral:
Mustafa Sentop, the parliament speaker and a close ally of Erdoğan,
waited for nearly two weeks to sign the referral of the protocol to
the relevant commissions for review and debate. Even after Sentop’s
signature, his office waited for another week to actually send the
protocol to the commissions, which finally happened on May 31, 2019.
The article notes that the delay is usually one or two days, not three weeks. Furthermore, it accuses the parliamentary committees of refusing to take the proposed protocol up for debate, wasting another few weeks.
If we assume that all the relevant players cooperate, the process could be completed relatively quickly - perhaps even within the same day - but realistically I think the process could be completed within a week. There is one final snag though: once the protocol is ratified it needs to be deposited with the United States. In 2019, this took over four months from the aforementioned presidential assent and was only completed on December 9th. This process, however, could also be completed very quickly if the political will was there.
Technically then, it's possible, however it's impossible to predict the future. With the elections, and now the earthquakes in Turkey & Syria to contend with, it seems very possible that the ratification of the protocol will slip past July 1st.