No, every living organism was created over the entire earth before the garden of Eden was even planted. The garden of Eden was not planted until thousands of years after all life, except that of humans. The creation of the universe and the earth was complete when described in the first verse of the first chapter of Genesis. Verse two focuses on how God put and molded the earth together. Verse three describes the beginning of God preparing, the already existing, earth for human habitation. Plants and animals were created and existed long before the garden of Eden was even planted.
Basic format of the Genesis account:
Genesis 1:1 God completely creates the universe and the earth within it.
Genesis 1:2 God focuses on how he molded the earth into a planet.
Genesis 1:3-2:4 God focuses how he prepared the, already existing, earth for human habitation. This includes, a general mention of his creation of the first man and woman.
Genesis 2:5-8 God focuses on how he prepared a garden in which the first human couple was to live and how he created the first man.
Genesis 2:9-14 God focuses on the water features of Eden and the garden planted within its boundaries.
Genesis 2:21-24 God focuses on how he created the first woman.