I have a 1989 Lennex gas furnace and a new (7 month) carrier comfort series heat pump.

In the mornings (thermostat is set for 06:00, temp 68) when the heat is to increase 4 degrees (from 64 to 68) the gas furnace goes on to requested temp. Then during the day the heat pump works. Is this normal?

The thermostat is set for the heat pump.

Heat pumps have poor recovery. They can't raise the temperature very fast. Generally, the thermostat would have to have some smarts to dictate furnace behavior. Some ways are:
1. Large difference between setpoint (gas heat)
2. HP struggling, no real change in temp per time (gas heat)
3. Adaptive recovery. System moves the time earlier to keep the gas from coming on.
4. Forcing the gas heat off based on outside temperature
5. Thermostat not really designed for dual fuel.

Electric resistance heat backup can be on at the same time as a HP, so the outside temp determines the range allowed.

Some may be thermostat options and some may require additional parts.

So its normal that the furnace goes on in the morning.

Yes, because your trying to raise the temperature quickly.

If you can turn it off emg heat in the stat then it would not.

If you had a sensor installed outside and the right stat, you could lock the gas heat out.

But, it depends a lot on the stat. What model stat?

The unit that came with it.

Edge programmable Carrier.
Edge Programmable Thermostat - Carrier

Normally a heat pump stat will bring on the aux heat if the temperature increase is higher then 2 degrees about ambient temperature:D

They mention a mode in the instructions which you need to contact the dealer to adjust settings. That would likely prevent the gas heat from coming on.

Thank you very much.

I will do that.