Brook is currently playing in the first of two Tests at Chesterfield, a series in which Middlesex batsman Max Holden is leading the side.
Yorkshire’s 18-year-old batsman will then take over the leadership duties for five one-day matches between August 7 and 16.
The right-hander has made an encouraging start to life in Yorkshire’s first team, scoring 38 and 31 in four innings in the County Championship.
His 38 on debut against Middlesex at Lord’s last month, following an excellent run of form in second-team cricket, was particularly impressive.
“Harry’s shown a lot of promise, hasn’t he?” said Moxon, Yorkshire’s director of cricket.
“His first innings at Lord’s was really good, and I think he showed then he’s got huge potential.
“He hasn’t quite got beyond his thirties yet, but, hopefully, that will come soon.
“He’s got a very good temperament, a good technique, and we are hoping he will go on and be an outstanding cricketer for us across many years.
“He just needs to play now and gain experience.”
Brook has captained Yorkshire’s Academy and in schools cricket. When asked whether he has the attributes to make a successful captain in the future, Moxon said: “I think he could captain at a higher level one day, yes.
“He’s not done that much of it yet, so it will be interesting to see how he goes with the Under-19s.
“He’s got a good cricket brain, and I’m sure he will do a good job. He will learn very quickly, which is what the best players do.
“His knowledge and cricket brain at a young age is such that he will learn very quickly and gain the experience needed to push on.
“It’s not always going to go right for him,” added Moxon, who confirmed the hand injury Brook suffered in the Championship defeat against Somerset at Scarborough earlier this month has all but healed.
Keighley-born Brook will miss Yorkshire’s next Championship match against Essex at Scarborough, starting on August 6, as a result of his England commitments.