ALTERNATER

Replace the battery first with a new one , at least to test it, if battery is not fully charging it may be the cause. After check that the alternator belt is well in place, with a good grip and is not squealing. If not you may fix it just replacing the belt or putting it back in tension.
More likely it is a problem with the alternator.
There is a factory recall on 2005 Murano alternator, if this is your model contact Nissan, as you may be entitled to a free repair job at your local concessionary.
Check here:
Auto Recalls: 2005 NISSAN MURANO - ELECTRICAL SYSTEM:ALTERNATOR ...
If not you will have to replace the alternator, in this case I suggest contacting a qualified mechanic.
Here you can buy the part

Regards

Check your charging system