Do we need any kind of protection for AC relay coil?

It's not unusual to switch one relay with the contacts of another with no "protection". You can add an RC snubber (typically something like 100 ohms in series with a few tenths of a microfarad) across the contact or across the coil to reduce EMI and sparking.

Such a snubber can also be useful across an inductive load. Or across the coil to prevent high dv/dt at turn-off from preventing commutation of a thyristor (if you are using an SSR, it will often be built into the SSR).

Sometimes MOVs (Metal Oxide Varistors) are used, but I would suggest not using them for such repetitive transients as they actually wear out over time and still allow rather high voltages and don't reduce dv/dt signficantly. If you must use them, take into account their eventual failure mode (usually shorted, followed by a certain amount of flames and smoke and then opening up, if they are across the power).

You don't need one if you turn off the AC relay with a SCR or triac, because those turn off at current zero automatically. In that case, there isn't any voltage induced at turnoff.

If you turn off the AC relay by another contact, you place an RC network, a snubber, over that contact, so the current isn't turned off immediately, but slowly reduced to the much lower value of the RCL series connection. That way the induced voltage is low and will not arc over the opening contact.

Instead of using a snubber you can also use a varistor. As long as the relay is energized the varistor has a very high resistance. If you switch of the relay the voltage created by the self-induction of the coil rises to a higher level than the supply voltage and the varistor becomes conductive.

The varistor must be selected by the supply voltage of your coil and the energy stored in your coil.

You can calculate the energy of your coil by: \$ E=\frac{1}{2}LI^2 \$