I will be running my first game next week, and it will consist of one character using Edge of the Empire and two using Force and Destiny. I had a good grasp on how Obligation works, but with the mixed group, I'm a little unsure. I know the starting Obligation is 15 (per player and not taken in account for any +'s for XP/Credits) for a three party group, but I wouldn't include the characters that do not have Obligation. So would it look like this:

EotE Character: 1-15
FaD Character 1: -

FaD Character 2: -

I would roll my d100, and if a 12 came up, then the Obligation would be triggered, and everyone would take the strain penalty.

Can someone please confirm this for me?

Sure. The thing is that although they don't start with obligation they can certainly take on obligation throughout the adventures either individually or as a group.

And if you roll doubles, which would only be an 11 in your group's case, the Strain Threshold hit is doubled.

Thank you for confirming this for me. I'm hoping my session goes well.

ALWAYS decide what game you are running first. Base your mechanics off of there and then allow the other books. Just because someone makes a Mystic, doesn't make them a F&D character. If you will be focussing on an Edge campaign (which I assume you are considering the forum), then everyone gets Obligation, no matter what book they got their Specialization from. Again, assuming Edge is the setting you are using.

You can mix and match but it gets complicated and I would never ever recommend it. as it tends to bog down the system.
For example, I ran a game that started as an EoTE, but we integrated the Duty and Contribution rules to show the progress of the group as they made their way up the ranks of a major criminal family and tossed Obligation as it no longer had a very impactful story presence.

That said, it's really easy to throw the Morality mechanic into AoR or EoTE even if for only one player who you want it to mean more to without many issues since the downside/benefit is on a mainly personal level and the story trigger is quite rare (1% chance). However, Obligation and Duty both affect the entire party and rolling for one or the other only for one character turns it into a meaningless task or a stealing the show type scenario since that's the only character that will trigger the mechanic.

Keep in mind that Obligation doesn't need to be seedy and dark as well. It can easily fit into your F&D games (if you are running that style instead). Such as the older mentor of the group having a strong Obligation to strictly following the Jedi code and his younger friend who looks up to him the Responsibility obligation with a connection to the mentor completing their bond.

TL;DR version:
Make it fun and integrated for everyone if you use the mechanic at all. Don't just throw it on one guy because he used an EoTE book while the others used F&D. If their moral choices are the main focus of your story, use Morality. If you want the group to have to wade into the criminal underworld in fringe space without getting drowned then Obligation is a good mechanic. Want your group to work their way up the ranks of some organization or fight for a cause, there's your Duty.

What OfficerZan said is what my group does as well. We pick which mechanic to use (Obligation/Duty/Morality) and then let players pick careers from any book. That way things don't get weird or confusing over who is getting what bonus.