Traditionally Solomon four group comes under experimental design. But there are many quasi-experimental studies conducted. Is there any book reference saying that it can come under quasi-experimental studies?
The reason that the Solomon four-group design is listed under "true experimental" (vs. quasi-experimental) designs is that randomization to condition was considered an integral part of the design.
If you're asking, could one contrive a 2 x 2 arrangement of existing/intact groups to draw inferences about the influence of two separate IVs, the answer is "Yes." It just wouldn't offer the same degree of confidence that the results were strictly a function of the IVs under investigation.
As well, one of the two IVs could be whether the group was pretested or not.
Although Campbell and Stanley did present the merits of the Solomon four-group design, it is rarely used in practice. Instead, the most common way of dealing with threats to vanity uses a 2x2 design, consisting of a pre- post-comparison with an experimental and a control group. This can indeed be implemented in a quasi-experimental format.