I want to convey to a client that if they don't take things seriously and provide incorrect information then our management will come back very hard on us.is this the right sentence or any idiom or vocabulary i can use in business english context
I think the idiom you're looking for is "come down very hard on us," defined as:
to express one's unfavorable opinion of the worth or quality of
You could also say that it will "come back to bite us"—in order words, it will cause problems later.
Rather than looking for an idiom, be specific about what the consequences are. To my ear, "come back hard on us" sounds like a veiled threat of some form of potentially illegal reprisal. Also, "us" is unclear - is it only the team of workers involved on your side, or does "us" include the client. If the consequences are entirely to your team, are you trying to make an appeal to the client's sense of fairness? It might be more effective to persuade them based on how their own concerns will be impacted by incorrect information rather than how your team will be impacted.