was there any government university there before this?
No.
Florida International University
In 1943, state senator Ernest 'Cap' Graham (father of future Florida governor and U.S. senator Bob Graham) presented the state legislature with the initial proposal for the establishment of a public university in South Florida. While his bill did not pass, Graham persisted in presenting his proposal to colleagues, advising them of Miami's need for a state university. He felt the establishment of a public university was necessary to serve the city's growing population.

In 1964, Senate Bill 711 was introduced by Florida senator Robert M. Haverfield. It instructed the state Board of Education and the Board of Regents (BOR), to begin planning for the development of a state university in Miami. The bill was signed into law by then-governor W. Haydon Burns in June 1965, marking Florida International University's official founding.
what part of florida is florida international university University Park, Miami. when was this university established? 1965. by whom? W. Haydon Burns, the Board of Education and the Board of Regents.
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Florida International University [SEP] History
In 1943, state senator Ernest 'Cap' Graham (father of future Florida governor and U.S. senator Bob Graham) presented the state legislature with the initial proposal for the establishment of a public university in South Florida. While his bill did not pass, Graham persisted in presenting his proposal to colleagues, advising them of Miami's need for a state university. He felt the establishment of a public university was necessary to serve the city's growing population. In 1964, Senate Bill 711 was introduced by Florida senator Robert M. Haverfield. It instructed the state Board of Education and the Board of Regents (BOR), to begin planning for the development of a state university in Miami.