Is it appropriate to say "To crack this hard nut"?

The following are the two common forms of the the idiom:

1. To crack the nut:

   Example: We have tried to crack the nut many times, but still have not found a design that consumers approve.

2. A hard nut to crack:

   Example: Fixing our relationship with the marketing department has been a hard nut crack.

(To crack this hard nut is not specifically a form that is generally familiar.)

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A similar idiom is a thorn in the side (the article before side may be substituted for a possessive pronoun). It generally means a problem that has created difficulties over a long period of time, often a conflict with other individuals or groups. For example, the oil lobby has long been the thorn in the side of the clean-air movement, or my injury from training has been a thorn in my side all season.

Finally, in regular speech, someone may say simply, "That's a tough one", which means essentially the problem you just explained is difficult to solve, or the question you just asked is difficult to answer in a helpful way. For example, "Can I buy food somewhere nearby?", "That's a tough one; most stores closed for the night".

You'd probably be understood, but I'd suggest "crack this nut" instead, leaving out "hard". Understand that in the original phrasing of the idiom, the adjective "hard/tough" is grammatically required because it has a dependent relationship with "to crack". In the rephrased version, though, the adjective isn't needed and becomes redundant—the fact that you're trying to "crack" this problem in the first place is enough to indicate that it's hard.

Also know that if you're talking about cracking nuts or anything else in the context of solving a problem, people will generally understand the allusion even if you don't adhere closely to the idiom.

If you want to say "to solve this difficult problem" then use that expression, "to solve this difficult problem" is clear and correct. You don't need to use a metaphor.

The metaphor is easy to understand, though it would be more common to say "(the problem) is a hard nut to crack". I just don't see any real benefit in using the metaphor.