The generalized Hosoya triangle is an arrangement of numbers in which each entry is a product of two generalized Fibonacci numbers. We prove the GCD property for the star of David of length two. We give necessary and sufficient conditions such that the star of David of length three satisfies the GCD property. We propose some open questions and a conjecture for the star of David of length bigger than or equal to four. We also study GCD properties and modularity properties of generalized Fibonacci numbers.