There is no official record of evictions in Canada. However, evictions that make it as far as a provincial tenancy board would be recorded. In some provinces, like Ontario, a landlord could theoretically access some previous eviction decisions from the LTB.
Or, if you’re evicted for missed rent, your landlord could report you to a credit bureau like Equifax or TransUnion. If they do, that missed payment would likely remain on your credit report for six years.