It's going to depend on the fiber architecture involved. For thin lamina with unidirectional fibers, you can use the formulas in "Mechanics of Composite Materials" by Robert M. Jones. However, these will only give you the in-plane engineering constants, namely E1, E2, G12 and nu12. For woven fibers, and especially 3D architectures, you will need to find other formulas.

Also for the case of unidirectional fibers, if you assume that a ply is "transversely isotropic" as discussed by Jones, you can get some of the out-of-plane constants in relation to the in-plane ones, namely E3 = E2, G13 = G12, and nu13 = nu12.

However, that leaves G23 and nu23 unknown. If you do not have test data for your material, you may have to make assumptions as to what these values are. Based on test data from similar materials, you may be able to assume a ratio of the known to the unknown values. In the case of unidirectional lamina for example, n23 might be in the range 1.0 to 1.7 times n12, and G23 might be in the range 0.3 to 0.7 times G12. These suggestions may or may not apply to your case. They might be good enough to get started, but if your problem is really 3D, you need to do more research into the actual properties rather than making assumptions such as I have described.

Oh thank you so much. This is informative! I really appreciate it!!