The chapter is a bit confusing, but from what I can understand, it seems like the noble was checking if Rizel was lying. Rizel seemed to do thing untypical of normal adventurer, so the noble was doubtful. He asked probing questions and then checked Rizel’s pulse. If Rizel lied, then his pulse would be faster and the noble would know then. Hence Jill’s comment that Rizel looked like he could manipulate his pulse, which meant that Jill thought Rizel could lie without a change in his heartbeat.
And about the argument, I think it has been one month since the start of Rizel and Jill’s contract. Rizel tried to pay the contract fee for one month of Jill’s service that he proposed in the first chapter plus some extra money. This is where it get a little confusing, but I think Rizel tried to let Jill go, only asking Jill to help him when something comes up by offering new contracts, thus treating their relationship like a business matter. However, Jill was hurt because he didn’t stay for the money and he thought that he was a “friend” (I put that in an air quote, since the shounen-ai vibe is really strong here. Seriously, see the interlocking hands and tell me that is a normal friend interaction.) Jill was pissed that Rizel tried to end things and treating the relationship like a business matter. Rizel claimed that he wanted Jill to be free and didn’t want to tie Jill down by being with him, but Jill wanted to stay nevertheless, even without the contract, because he found it interesting.