The short answer is no. Apartment complexes are real estate investments there to generate income and jobs for property owners. You can be sure that there are safety measures in place to protect that investment.
The standard credit check process, the background checks, and, in some cases, the reference checks will eventually expose the truth of your income.
If you are applying for federally subsidized housing, your income determines your rate of rent. So you will need to provide information on your income by showing recent pay stubs, income letters from Social Security, and current bank statements. Falsifying this information can put you at risk for eviction, loss of benefits, and prosecution by state and federal agencies. The documents you provide to qualify for housing subsidies have to be submitted yearly and are scrutinized by the auditing agencies. Discrepancies by even one dollar can cause your benefits to be red-flagged or event rejected.
Honesty is always best in any aspect of life.