Hi esconda,
The Galileo supports the mPCIe card and Bluetooth shields. The Bluetooth shields are Arduino compatible and can be placed in the Galileo, so Arduino software will work as well. In this link you will find the tested shields of the Galileo Gen 1and Gen2 and see that the HC-06 is in fact tested https://communities.intel.com/docs/DOC-22996 . Take a look at the 2 shields using the HC-06 in section 5 and 25. There you will find some example codes; they might be useful for you.
Regards
Sergio