In my ASL beginners course I taught on the first day of class in the early 2010s, the fresh-blood students in small groups discussed about what they had learned from the ASL/Deaf Awareness quiz. Across the room, I "overheard" (via an interpreter) when a student told his group that there is no accent in sign language. I intervened that accents do exist in sign language. The student and other students were surprised that 1) There are accents in sign language, and 2) I "overhead" or caught him and responded to him in ASL via an interpreter.
Does a signer have an accent in their signed language? Yes, definitely! Everyone has an accent. An accent is a manner of pronunciation of a language. It is the way one pronounces a language.
A person's accent is developed as she/he learns the language where she/he is from or what social groups they belong to. There are other demographics involved, such as age or generation, social identity, education, etc. How signers pronounce or utter can convey some characteristics of their signing.
An accent can give the impressions about the speaker but it depends on who the listener is and how he/she knows about the impressions of the accent. For example, Deaf native-signers, especially those from native-signing Deaf families, can finely detect whether a signer is Deaf or hearing, native or non-native, oral deaf or culturally Deaf.
Not only audiologically but also geographically, native signers from different regions have accents. E.g. Deaf New Yorkers.
A native accent is the way one speaks in their native language. With the foreign accent, a speaker uses his/her manner or style of their native language into another language.
Accents can change over time when language changes. Some signing Deaf immigrants change their accents over many years while some other signing Deaf immigrants maintain their accents of their native language. It depends on where they socialize and evolve in a socio-lingual group.
Interestingly, even a Deaf child of the native-signing Deaf family (who never vocally speak) had a vocal accent from an European country where she had a rudimentary speech therapy at a preschool school for the deaf before her family emigrated to North America in the 1980s. The speech therapist was puzzled why the Deaf child sounded differently from other deaf kids she worked with at the school for the deaf. She couldn't figure out until one day she visited the family's home, where, aha moment, she learned that the child just emigrated to North America. Furthermore, that kid, who never used speech in her daily life other than the school's required speech therapy sessions, still had her foreign vocal accent over many years while she gradually lost her foreign accent in ASL.
Hearing signers, even fluent ones, and interpreters have a certain accent that Deaf native-signers can detect whether they are hearing or not. To native eye of Deaf native-signers and Deaf family members, a hearing signer passes as a native signer within a prolonged period of time is very rare. It is also true that some deaf signers, who don't grow up in Deaf world or bilingual deaf schools as most of them these days are mainstreamed with hearing interpreters, have accents.
In the late 2000s, a hearing signer befriended with a Deaf native-signer and let her know that he hadn't signed in a long time and would need a refresher. She told him that she recognized his signing from the 1980s. Sure enough, he learned ASL in the 1980s and later hadn't used it for a couple of decades. In addition to his hearing accent, he also had accents of the time period and the region.
Aside from accents, other factors also play into one's overal signing appearance. E.g. Personalities such as laid-back personality or feisty personality. Old age or young age. Emotions and moods of the moments. Ethnicities. Genders. Combinations of those factors create endless gradients of signing styles.