The big question is: how prominent are the vikings in your Arabian-Muslim setting and how prominent is their culture compared to the Arab-Muslim province/state/empire they are living in.
If we are talking about just a handful of individuals, may be a few trader families or a band of mercenaries settling in a majority Arab state, they will probably just keep their names, but adjust them somehow for Arab spelling. As most of the surnames are patronyms, they might just replace the Scandinavian '-son' with the Arab 'ben' or 'bin'. So Harold Björnson would become Arruld Bin Bijerni. Eventually, they will intermarry with the local population and though at first their offspring will have some 'noncommittal' name that goes well in both Scandinavian and Arab culture such as Achim or Leia. One or two generations later however their ancestry will be largely forgotten and their names will just be whatever is popular in their hometown at that time. So Harold Björnson will become Arruld bin Bijerni. His son will be Al-Frid bin Arruld and his grandson Harun bin Al-Frid. His grand-grandson will have a completely Arab name like Nour bin Harun, but might well be nicknamed Nour the red-haired.
If instead we have a largely self-sufficient community adopting Islam and becoming part of an Arab state, remnants of their culture will remain in their names as well as in their customs. So like in the first scenario, their names will just be adjusted for spelling. We might even see the ending -son stick around in local use, possibly even in official writings as it is now a part of the local Muslim community. Eventually we will see religion seep into the namegiving. Names based on Norse gods such as Thorstig will fall out of fashion and be replaced by names of Islam significance like Muhammed, Bilal or Hassan. (Much like all the Christians and Christinas in Sweden today.) However a larger influence will come from the overlaying culture and fashion and if a name like Suleiman suddenly becomes popular all over the country, their will inevitably be lots of children with that name in the Viking-Arab regions as well. Kind-of like we can today find girls named Ashley in the US as well as Sweden, Kenya, Brazil and Hong-Kong.
(Of course, the opposite can happen as well. A name from the Viking-Muslim towns can become so popular that is will be adopted all over the country and eventually all over the Muslim world. So prepare for some Sulaveig a Mursi in Egypt, named after the famous Viking-Muslim actress Sulaveig al Harunson.)
If the community is large enough, about state-sized, names and culture will largely remain the same at first, but eventually there will be a new culture emerging from mixing the Viking and Arab heritages. So like we now have names like Khemal and Tajif in Turkey, we might end up with names like Stiig and Gudwin in Viking-Arabia.