Social Security Disability Insurance (SSDI) is a benefit that is earned based on how long the person worked and what their income was. SSDI is not affected by need or other support.
Supplemental Security Income (SSI) is needs-based. The people on this forum are tax experts, not SSI experts, so you might get a better opinion by talking to a social worker or contacting the social security administration. Certainly if you claim the person as a dependent on your tax return, the fact they are claimed could be sent by the IRS to the social security administration--but I don't know if it actually is.
In any case, if they are living in your home, that fact should be disclosed to SSA even if you don't claim them, if you support this person and your support is not reported to the system and included in their calculations of need, that could be considered fraud whether or not you claim them as a dependent. Whether it changes their payment will depend on your overall financial situation (other dependents, other expenses, income, and so on).
And finally, if you marry this person, that may change the needs calculation as well, so you might as well tell the SSA the truth sooner rather than later.
