Since your mother gifted it to you through a quit claim deed, you would claim the method of acquiring it as a gift with date of acquisition as their purchase date, and the cost basis the same as their cost basis plus any costs of improvements over the years.

Since you meet the requirement for use of the house as a primary residence or 2 out of the 5 years prior to sale, if you are single, up to $250,000 of any gain would be exempt from tax, leaving very little in taxable gain at the long-term capital gains rate.  By time you subtract selling costs from your sale price, you would likely pay no tax.