Function Generator Voltage and Current handling
No.  This will burn out your MOSFET as there is nothing to limit the current from drain to source.  This should not harm your function generator, since the MOSFET's gate will not draw current other than to satisfy its capacitance.
This unit has a 50 Ω output capable of ±20 V into high impedance or open circuit (±10 V if the frequency is > 25 MHz).  Into a 50 Ω load it can do ±10 V (±5 V if the frequency is > 25 MHz).  You can use Ohm's Law to derive the currents if that is of interest to you.
Output impedance and voltage range is listed on page 4 of your datasheet under "General Characteristics - Amplitude".