This conversation is from "The BigBang Theory" Season 12 Episode 2.
Around 25 sec, the character says "I'm gonna have to say 'yes'."
I haven't seen "be going to..." and "have to" used in such way before. In my understanding, "be going to..." indicates a future plan or a future event.
"I'm going to visit my antie in Vienna next holiday." "One day I’m gonna be a star." "Look, the sky is dark. It’s going to snow again soon."
So maybe I don't decently get what he tries to convey here. Using "I'm gonna have to say...", not "I'm gonna say," "I have to say," "I'd say," "I'll say" etc, what is this line suggesting?

Have to = must / need / obligated to do something! 😊

You're 100% correct with the use of "gonna / going to" as an indicator of future actions. We use many words that have similar meanings for redundancy to double up and rrrreeeeeaaallyyyyy emphasize the concept we're trying to convey. "My crush just asked me out on a date? I'm gonna have to say yes, duh." You don't have to, you don't need to, say <gonna> here. The point that you "have to" say yes is clear enough when you say "I have to". Adding "gonna" intensifies that need, like I REALLY need to say yes.

For example, Howard saying "I'm <gonna> have to say yes" can be replaced with "I <really> have to say yes". As in, "I really need to tell you it's a scavenger hunt even though it's not just so I can watch you two running around the city searching for your next clue like absolute fools 😂" ahahahahaha

It can also be used to soften a tone, to make it more passive. For example, instead of "You're fired," or even "We have to let you go," "We're going to have to let you go" is even softer than either of those since it emphasizes even more that there's some sort of need to do it, rather than it being your fault for firing them or their poor performance as an employee. "It just has to be that way." Or if you don't want to go on a date with someone but don't want to tell them directly you can say, "Ah dang, my brother's in the hospital so I'm <gonna> have to cancel to visit him". "gonna have to" is even softer than plain "have to".

"Gee, I just got a raise and my favorite band is in town for one night and one night only...I know I said I was gonna start saving money, but with this once-in-a-lifetime opportunity coming up, I guess I'm just gonna have to go to that concert...! Really can't do anything about it...! Just gonna have to do it!" It puts greater emphasis on what you're saying so they know it really "is" out of your control...even it it's not! 😂 Whether it is the truth or you're lying, you're trying to convince them that it is, in fact, true! Ahaha! 😂