I know that volcanoes and earthquakes occur on plate margins. But, do some of them also occur within plates?
Crustal plates are not homogeneous, uniformly continuous rock masses with uniform stresses. They are a mixture of rock types with variable stresses and stress concentrations. They are also discontinuous.
There are fault zones within plates. Some faults are active & others inactive. When movement occurs along an active fault, usually due to in-plate localized stresses, an in-plate earthquake will occur.
Concerning in-plate volcanoes. These are due to geological hotspots. These are,
thought to be fed by underlying mantle that is anomalously hot compared with the surrounding mantle. Their position on the Earth's surface is independent of tectonic plate boundaries. There are two hypotheses that attempt to explain their origins. One suggests that hotspots are due to mantle plumes that rise as thermal diapirs from the core–mantle boundary. The other hypothesis is that lithospheric extension permits the passive rising of melt from shallow depths.
Intra-plate volcanism occurs when rifting of the plate occurs. But you don't necessarily need a hotspot (or rather a mantle plume) to trigger a rifting process. There are two types of rifting:
- Active rifting, which indeed implies the rise of a mantle plume under the lithosphere. This yields to the bulging of the lithosphere, then to its extension. In such systems, volcanism usually happens early, while sedimentation happens at a later stage. Example: the East African rift.
- Passive rifting. In this system, the extension of the lithosphere is triggered only by the stress at the plate boundaries. This extension allows the asthenosphere to rise passively. In this case the sedimentation in the basin happens before the volcanism. Example: the Rio Grande rift.
Note that nature does not comply with these idealized models! In French Massif Central, there is evidence for a two-stage rifting, first passive then active. Merle and Michon (2001) proposed a model where the formation of the Alps lithospheric root induces an asthenospheric flow, leading to this two-stage rifting with no proper mantle plume involved.