Euler graph is defined as:
If some closed walk in a graph contains all the edges of the graph then the walk is called an Euler line and the graph is called an Euler graph
Whereas a Unicursal Graph does an open walk. So can we say a unicursal graph is an Euler graph?
In the terminology of the Wikipedia article, unicursal and eulerian both refer to graphs admitting closed walks, and graphs that admit open walks are called traversable or semi-eulerian. So I'll avoid those terms in my answer.
Any graph that admits a closed walk also admits an open walk, because a closed walk is just an open walk with coinciding endpoints.
Vice versa, this is not always the case. Not any graph that admits an open walk also admits a closed walk. A simple example for this is a graph with only one edge.