How do I get Syndicate members to betray each other more often?

As the title implies, having betrayal members betray one another, causing them to be rivals, allows for Betrayal encounters to be more rewarding. The more members are rivals, the more Betrayal members you can encounter in a single division attack. However, it seems as of late I've been depending on RNG, or just not optimizing my Betrayal board, as it seems I am getting less of Betrayal options, and more stuff like:

• +# to division intelligence
• Drop a veiled weapon/unique weapon/currency/div card
• Swap positions with one another
• etc.

Essentially, what should I be doing to be maximizing the board so they no one is trusting one another? Do members need to trust each other before betraying each other?

Your intuition is correct: Syndicate members need to trust each other (green connection lines) before a betrayal option appears. Both green and red connections increase the average number of Syndicate members you can interrogate per one encounter. Because of that, you want to create both friendships and rivalries in your Syndicate.