Medicare Part A ( Hospital Insurance benefit portion) is the benefit that you have been paying for during your working years so there is no additional premium to pay for this part of Medicare.
The only reason NOT to sign up for Part A would be if you ( wife plan) have and are still contributing to an HSA and want to continue - HSA and Medicare do not work together.
When you sign up for Medicare, you will receive a card with your Medicare number and the card will identify what parts of the benefit you currently have signed up for with the date of initiation. The way I understand it, the company will issue you a notice stating that you have credible coverage for the other Parts of Medicare - Part B (doctors, etc.) and Part D (prescription drug coverage). Keep this in case you need it.
Part A can often work with the employers plan or a retiree plan - usually if there are more than 20 employees - the group plan pays 1st and then Medicare Part A would pick up anything, if anything, which they might cover under Part A.
This Medicare.gov pamphlet will further describe how they work together.
Medicare.gov - CMS - Medicare & Other Health Benefits - Your Guide To Who Pays First
(revised December 2018) page 12 seems to be what you are asking about.
Just remember that when you no longer have this group coverage under your wife's employer, you must pick up Part B / Part D and pay the monthly premiums. From what I understand, you should have documentation of when your credible coverage ended under the employer plan - the date is important because you only have a window to pick up Medicare Part B and Part D and if you miss that window, there is a penalty that can be attached to the monthly premium.
It's Always Something . . . . Roseanna Roseannadanna