New bathroom fan has 2 green wires but i do not know where to connect them

Without actually being there, and not being able to see the wiring at each of the devices, it's difficult to be sure what's going on. But here's my best guess, based on past experience.

To make it easier, I've numbered the ungrounded (hot) conductors in the photo from left to right.

1. You've already established goes to a light.
2. Supplies power to the switches.
3. Likely goes to the fan.
4. Also goes to the fan, but since the fan doesn't have a light built in, it's likely capped off at the fan.
5. Probably feeds some other device, maybe a receptacle.

Why 3 and 5 are controlled by the same switch, I have no idea. Again, I can't see the room, so I have no idea what equipment you have.

If 1 and 4 are both for lights, it makes sense that a singe switch controls them.

One other thing to check, is to make sure the terminals of those switches are rated to accept more than one conductor. There should be a label on the switch that tells you what type, and how many conductors can connect.

Verify with a voltmeter that the fan is indeed getting 120V AC. If it does, and there's no physical obstruction to the fan blades, bring it back to where you bought it, the motor is faulty.

If there's no 120V, you'll need to check your wiring, and the switch. Any kind of automatic fan switch (humidity sensor, timer, etc.) is especially suspect. Replace it with a normal on/off switch and see if that solves the issue.