It's the other way around. The characters deduce Vegeta is trying to get UI by the way he's fighting, and nobody even says Vegeta's power is rising until he starts to charge the Final Flash. When Jiren knocks him down the first time we even see Vegeta analyzing his moves as he fights Goku, and Piccolo even says as much when Vegeta lands his first blow.
Also if Jiren is using the same power as before then shouldn't Vegeta have unlocked UI right there? He's got the power and the fighting style and is even pushing beyond his limits when Jiren regains the upperhand.
First, let's look at the title of the episode. Episode 122 is titled "With his Pride on the Line! Vegeta's Challenge to Be the Strongest!" Pretty obvious what the entails. Vegeta's Saiyan pride is what's causing him to challenge Jiren. He challenges Jiren because he wants to be the strongest. By fighting Jiren, he will be the strongest by obtaining Ultra Instinct which is as Whis says. Whis makes that statement in response to everyone noticing how Vegeta is pushing back against Jiren even harder to the point where Jiren can no longer fight back.
After Vegeta breaches Jiren's lightning fast bullet-hell punches, Vegeta begins overwhelming Jiren and pushes him back. Here's what Belmod says in response to Vegeta overwhelming Jiren.
Episode: 122
Time: 12:46-12:52
Context: Vegeta begins pushing back Jiren.
Belmod: What's wrong, Jiren?!
Kaioshin: Is he... winning?
This is when Whis confirms that Vegeta is attempting to obtain Ultra Instinct. Afterwards, we see Vegeta literally zip around Jiren and then zip around his line of sight again to land a critical blow. That's a clear demonstration of speed. It's at this moment where Vegeta begins to taunt Jiren.
Episode: 122
Time: 13:30-13:34
Context: Vegeta lands another critical blow on Jiren.
Vegeta: You were faster and stronger when you fought Kakarot!"
This goes against Vegeta's character. Normally he's filled with confidence but he knows Jiren isn't using his full power. He's pressuring Jiren so that he can push himself even further beyond his limits to obtain Ultra Instinct which is why Whis' statement is made under that context. It's not just a reference to that specific moment in time but during the fight as a whole.
Still not convinced? Let's take another look after Vegeta recovers against Jiren.
Episode: 123
Time: 18:52-19:06
Context: Vegeta begins powering up with Goku against Jiren.
Vegeta: I have to keep my promise to him. That's why I'll surpass my limits in my own way! And, Jiren, I'll beat you!
Vegeta makes it a clear point that he wants to beat Jiren as early as Episode 122. He does so by trying to obtain Ultra Instinct which he knows from watching Goku fight is a limit-shattering ability. Once Vegeta realizes it's not for him, he attempts to break through his limits his own way. The concept of pushing himself to his limits was always the case the moment he challenged Jiren. He wasn't satisfied with the level he was at hence why Vegeta taunts Jiren by stating that he was fighting harder against Kakarot. Vegeta's pride is what compels Vegeta to keep pushing forward. Everyone takes notice of that. Belmod is concerned. Blue Evolved is a culmination in Vegeta's desire to be the strongest by pursuing another path once he realizes Ultra Instinct is not it.
I think the message is clear that Vegeta was growing stronger through his pride this entire time. It's even implied in the Episode title.