I are using nrf52832 to develop a product that needs FCC certificate,
We use a PA and can increase the output power up to 20 dBm and it
seems that is lower than FCC limitations (its 21 dBm).

In FCC 15.247-2 it is said "The minimum 6 dB bandwidth
shall be at least 500 kHz". I think it is OK for 52832.

But in another part of FCC they defined the limitation of the power density
as bellow:
"For digitally modulated systems,
the power spectral density conducted
from the intentional radiator to the
antenna shall not be greater than 8
dBm in any 3 kHz band during any time
interval of continuous transmission."

My question is that how much we can give power to antenna without closing
to above mentioned limitation about power density.
One of my friends told me that sometime a Nordic Specialist said for NRF52832
the maximum power level that can give the antenna after the PA without going over
the limitations of power density (8dBm for 3KHz) is 13 dBm. Is it correct and can you give
me your calculations your help me to find it?

You really need a specialist to guide you. If you are asking this type of question there are 100 other things you are going to fail on during certification.

Not sure where you got the 21dBm number from, BLE is not frequency hopping by FCC standards. The channelization is wrong, the hop rate is wrong. BLE is DTS so max power is 1 watt.

On PSD, easiest approach is to just measure it and gain up from whatever you measure. Luckily BLE uses GFSK and not pure FSK. The Gaussian filter spreads out the carriers a bit. A stock 52832 at 4dBm comes in at about -20dBm for 3kHz RBW so I'll let you do the math and guess what you can drive the conducted power output to.

Your biggest problem will be the restricted bands. 2.4GHz is surrounded by restricted bands and the harmonics fall into restricted bands. I would recommend you start by figuring out what your power output can be while still passing restricted band measurements.

I should also point out that you will never be able to ship your proposed product into ETSI communities. The way the PSD spec is written for ETSI (aka, RED) the max power output is 10dBm.