No. This is not easily accomplished in Inkscape.
The closest work around would be to extend the two lines, and using snapping to draw a new path over them.
- Using the Node Tool F2, select the nodes to be extended.
- Hold Ctrl + Alt and drag the node to extend it. Repeat with the other line until they intersect.
- Turn on snapping. Make sure that "Snap to Intersections" is enabled.
- Draw a new line Shift + F6, using the end points and intersection of your two lines as snapping points. (If Inkscape doesn't automatically snap to these, check what snapping options you have set and try again).
- Delete the original two lines
If you're working with lines that are right angles, it might be easy to draw a single line between two end points, add a node in the middle, and then manually assign coordinates to that node instead of drawing a new line.
This operation isn't too difficult if you have 2 simple straight lines. There's a good chance that your drawing is more complex than this and you'll need to integrate this with other objects. The commands in the Path menu, such as Union can help with those finishing touches.
Inkscape can be extended with scripts. It might be possible to write a script that will do this all for you at the click of a button, as you wanted, however I'm not aware of any.