According to the University of Nottingham, Nottingham General Hospital was
founded as a charitable institution by public subscription in 1782
which suggests that patients did not need to fund their own care in the 1830s if they were judged as deserving.
(The concepts of the deserving and undeserving poor -- and also the impotent poor -- crop up in many discussions of poverty in the England from the 16th century onwards. The undeserving or idle poor were capable of work but would not. The deserving poor were those who would work but could not temporarily due to unemployment, accident, or ill-health. The impotent poor were those too young, too old or permanently too unwell to work.)
Unfortunately, it looks as if the records (accounts, minutes, annual reports) that survive from your period of interest are unlikely to shed light on your ancestors, unless there was something that led them to be mentioned in the accounts or minutes. However, the records would still be worth consulting to understand how the hospital operated/was funded and the types of patients who were admitted.
Nottingham Hospital History casts some light on the hoops that had to be gone through to get admitted:
"No patient can be admitted (except in cases of accidents) without a
recommendatory letters signed by a subscriber", and recommendations
were in proportion to subscriptions given. For one hundred pounds
down or five guineas annually a subscriber could recommend " six
in-and twenty outpatients annually but shall not held more than two at
any time in the house." However, subscribers her duties as well as
rights and "in the case of death the person who recommended the
deceased must either remove the corpse or defray the expenses."
To get into hospital the "object of distress" had to present himself
at the hospital on Tuesday with a letter of recommendation. After
examination by the physicians and surgeons, he was then questioned by
the Weekly Board to ensure that he was "a proper object of charity."