There are two variations of this:
If there are any other issues, please contact our customer support team.
If there are any other issues please contact Customer Support.
The first one is using customer support as a common noun, while the second one is using it as a proper noun.
One quick way of determining which to use (although it doesn't apply to every situation) is to replace customer support with another proper name. Let's use Mary.
If the sentence looks wrong with Mary, then use the lowercase noun; if it looks okay with Mary, then you can use the uppercase noun:
✘ If there are any other issues, please contact our Mary team. [Use customer support.]
✔ If there are any other issues, please contact Mary. [Use Customer Support.]