Yes, your understanding is correct. Gluons are indeed the force carrier particle of the strong nuclear force. This means that they mediate the interaction between quarks, which are the fundamental building blocks of protons and neutrons. Without gluons, quarks would not be able to bind together to form these larger particles.
To answer your second question, gluons hold together quarks to form protons and neutrons, which in turn make up the atomic nuclei. So in a way, gluons do indirectly hold together atomic nuclei by keeping the quarks bound together.
I hope this helps clarify your understanding of gluons and their role in the strong nuclear force. Keep exploring and learning about the fascinating world of particle physics!