FastQC will work fine, just remember that some of the modules will show the dataset as having failed.

At the fastq level, you cannot tell much I think, just the quality of the sequencing.

However, you can check some other metrics after mapping, such as the conversion rate (especially if you have a spike-in lambda genome). You can also check the mapped and un-mapped reads, the CpG and non-CpG site (generally non-CpG is very low except in neurons and embryonic steam cells).

if you use Bismark, you can check the summary reports. You can also use tools such as MethPipe to check the conversion rate.