You probably can use a regular PCR if you are just looking for presence/absence. Also if the viruses are common then you probably can find some papers with diagnostic primers. If you want to enrich for viral genomes it is recommended you pass the supernatant of your cell culture through a fine filter to remove cells before precipitating and extracting though this is usually only used if you are interested in sequencing. Still for PCR whole cell lysates should be sufficient. If it is an RNA virus you need to make cDNA after extraction.

There are special kits for isolation of viral DNA and/or RNA

Like this: https://base-asia.com/product/primeway-viral-dna-rna-extraction-kit/#:~:text=PrimeWay%20Viral%20DNA%20%2F%20RNA%20Extraction%20Kit%20is%20designed%20specifically%20for,and%20viral%20transport%20medium%20swabs.

https://www.thermofisher.com/am/en/home/life-science/dna-rna-purification-analysis/viral-rna-dna-purification.html

https://www.biotechrabbit.com/products/na-purification/virus-rna-and-dna-isolation/genuptm-virus-dna-rna-kit.html

Usually they work properly

Lyse the cells and release host cell DNA and RNA (simple water can burst host cells), then digest host DNA/RNA with nuclease such as benzonase, at this time the virus is still safe. And then you can isolate and purify the DNA or RNA from viruses for subsequent PCR.

For the detailed buffer and parameters, you may check publications by using keywords like: benzonase, virus, host cell, DNA, RNA, et al.