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The short answer is: yes, but it might prove quite challenging. The long answer is that you will have to set the Edison to pair without any Linux input (as discussed in https://communities.intel.com/message/292975#292975), set the Bluetooth interface to be ready on boot (something similar to https://communities.intel.com/docs/DOC-102152), set the Bluetooth profile (something similar to https://communities.intel.com/docs/DOC-100754 if you'd like to send the texts through SPP) and your own scripting.
This is an interesting project and I encourage you to try it and let us know your output. Some other documents that are probably going to are:
I can't connect from any app. I only can connect from settings (it is visible as a computer). I have my own app which is able to connect to arduino hc-05 but not to edison
[CHG] Device C0:EE:FB:43:9D:73 Connected: yes [CHG] Device C0:EE:FB:43:9D:73 Connected: no