Depends on the amounts involved.
I'm not sure what the threshold is for reporting payments through RTI on a payroll, but theoretically it would have to be processed through a payroll and be reported to HMRC even if no NIC/income tax due.
Used to be that people would pay an amount below the NIC threshold so no payroll necessary but I think RTI changed that. I'm sure someone with payroll expertise will appear.
If the amount is below NIC/income tax thresholds then whether or not the receiving spouse has to specifically declare it depends on whether they have other sources of income. E.g. If they completed a tax return it would have to be included on it to make sure their tax was calculated correctly. If they have another employment it would be to be taken into account in their tax code (e.g. If their personal allowance has already been used up by it).
If the business is a sole trade them it would appear in the accounts used to prepare their tax return but those are not public.
If they have a limited company the accounts would be on Companies House and available to the public but I would expect them to have filed filleted accounts without the profit and loss account where it would be.
So unless they have the spouse a copy of the accounts, no you couldn't really verify much. Especially if that person hasn't been running a payroll, as you wouldn't be able to ask HMRC what income has been declared for you.
If the person has put an amount through their accounts and claimed a tax deduction but not paid the money to their spouse (or processed a payroll or told the spouse) there are certainly questions to be asked. Spouse wouldn't have to sign anything for them to be able to do any of this. If running a payroll would need spouse's details like any employer but presumably they may easily have access to that regardless of involvement of spouse.