Is it illegal to not have tenant insurance in Quebec?

No, it isn't illegal. There's no law in Quebec requiring tenants to carry insurance, so going without isn't an offence. It can still put you in breach of your lease if your lease contains an insurance clause, and it doesn't reduce your legal responsibility for damage you cause. It only means you'd be paying that bill yourself instead of your insurer paying it.

What going without can cost you

If a fire starts in your kitchen or your bathtub overflows into the unit below, you can be held responsible for the damage. Without insurance, the landlord or the neighbour can claim those repair costs from you directly, and you'd also replace your own belongings out of pocket.

If your lease has an insurance clause

Letting your coverage lapse means you're not meeting an obligation of your lease. If that causes your landlord serious harm, they can apply to the Tribunal administratif du logement (TAL).

Tenant insurance is usually affordable compared to the risk. Talk to a broker to find coverage that fits your budget and your lease.