No. Spacetime is locally flat (Minkowskian). Same with the surface of a sphere. In order to detect curvature, you either have to move out of the manifold; which you can do by climbing a mountain on Earth, but you can't do in spacetime; or you need to take a survey. Read the introductory chapters of Spacetime Physics by Wheeler and Taylor, and, more importantly, of MTW's Gravitation.
But you ask a good question. The negative answer is the essence of Einstein's Equivalence Principle.
In response to comments, let me add that: by survey I really mean what those guys with site glasses and tripods do. It was how Gauss came up with his method of intrinsically determining curvature. The book itself is quite advanced, but the introduction explains how this relates to the equivalence principle without too much math, so I suggest reading the first chapter of Gravitation and Inertia The math in the second chapter gets a bit more intense, but the verbal part of the discussion of the equivalence principle stands pretty much on its own.
The seminal thought experiment is Einstein's elevator. If you were in deep space, confined to a windowless elevator, you would not be able to do a local experiment to prove that you were not freely falling in a "gravitational field". Just as, if you are standing on the shore line, you will not be able to see the curvature of Earth. (within certain limits of precision).
The larger the elevator, the more likely you will be able to detect tidal forces (curvature,) because you can measure the relative acceleration of more distantly separated test objects.
Imagine you were on a perfectly smooth planetary surface, and you wanted to paint rectangular parking spaces. If you restrict your parking lot to a few hundred meters, you will get away with using a square and some string to map out the lot. If you want to do the same over several miles, you will eventually run into problems because lines which are parallel at on location will become closer at other locations.