This runs into the problem of most threads about 'What if Country X joined the Axis?' Yes, there are several ways that Turkey could help the Axis position. On the other hand, it's very hard to see what the Turks could gain from it, and more to the point how their government could possibly change to the extent they'd be interested.
The Ottomans were dead, overthrown by the nationalists running Turkey in WWII- they don't want the empire's territories.
Enver Pasha is dead- the Kemalists are not pan Turkists.
The only thing Turkish entry into the Axis would gain them is a war with literally all their powerful neighbours. If the Axis was clearly winning- the Russians back to the Urals, the Med closed off- then the Turks would jump in to preserve national sovereignty like they did OTL.
Otherwise, forget it.