You can put the code you want to execute in the onRowAdd callback function, for example

    <MaterialTable
      title="Timesheet"
      columns={columns}
      data={DataArray}
      icons={tableIcons}
      editable={{
        onRowAddCancelled: rowData => console.log('Row adding cancelled'),
        onRowUpdateCancelled: rowData => console.log('Row editing cancelled'),
        onRowAdd: newData => {
            console.log('Hello');
            return new Promise((resolve, reject) => {
                setTimeout(() => {
                    // add row
                }, 1000);
            });  
        }
      }}
      options={{
        actionsColumnIndex: -1,
        addRowPosition: "first",
        search:false,
        exportButton: true,
      }}
    />
  );

If you want to call your function on the + button click, you need to add the actions. For example

    <MaterialTable      
      actions={[
        {
          icon: 'add',
          tooltip: 'Add User',
          isFreeAction: true,
          onClick: (event) => alert("You want to add a new row")
        }
      ]}
    />