You are not eligible to contribute to an HSA for any month that you are covered by any other insurance, including Medicare. I suppose you do not have to enroll in Medicare when you turn 65, but most people do because there are penalties for enrolling late. Assuming that you did enroll in Medicare, then your HSA contributions for 2020 are limited. If you had single coverage, your annual maximum is $3550, plus $1000 catch up provision for being over age 55. That works out to $379.16 for each month that you were enrolled in an eligible HDHP with no other healthcare coverage. If you were enrolled in a family HDHP, then your annual maximum contribution is $7100 plus $1000 catch-up contribution, or $675 per month of eligibility. Eligibility is determined by what kind of insurance coverage you have on the first day of each month.
Also, if you were only enrolled in a qualifying HDHP in 2019 for part of the year, but made maximum contributions using the last month rule, you will owe an additional penalty for excess contributions since you do not meet the test to use the last month rule for 2019 if you enrolled in Medicare during 2020.
