He is feeling fine and we do not expect any problems with his health this time.

Il se sent bien et nous ne comptons pas d'avoir (à avoir somehow doesn't sound good) des problèmes avec sa santé cette fois.
This is how I put it; the translator site said:
......nous ne comptons pas sur n'avoir aucun problème cette fois avec sa santé.
Which is correct and is there maybe a better way of saying this?
Merci beaucoup!

Merci beaucoup Mme; I truly appreciate the excellent help you are always providing. I will bookmark that translating site.

Yes, I did mean "this time" (not like last time, when there were some health problems).
Have a wonderful day!

this time = cette fois

You got it right!

Both translations you provided are fairly accurate. However, there are slight differences in word choice and phrasing that can affect the overall meaning.

Your translation: "Il se sent bien et nous ne comptons pas d'avoir des problèmes avec sa santé cette fois."
This translation conveys the intended meaning effectively. The phrase "ne comptons pas d'avoir" means "do not expect to have" or "do not foresee having." So, in this sentence, it emphasizes that you do not anticipate any health problems this time.

The translation from the website: "Nous ne comptons pas sur n'avoir aucun problème cette fois avec sa santé."
This translation also conveys the same meaning, but there are some subtle differences in phrasing. The phrase "ne comptons pas sur" means "do not rely on" or "do not count on." The phrase "n'avoir aucun problème" means "to have no problems." So, this phrasing emphasizes that you are not relying on or counting on having any problems with his health this time.

Both translations are correct, but I would suggest a slightly different alternative that maintains the meaning while using different phrasing:

"Il se sent bien et nous ne prévoyons aucun problème de santé cette fois-ci."
This translates to "He is feeling fine, and we are not anticipating any health problems this time." It conveys the same idea as the previous translations, but the word choice and structure may be more commonly used.

Ultimately, choosing which translation to use depends on your preference and the context in which you're communicating.