It seems very unlikely. Why would Lincoln arrest men who, even if critical of the government, were even more critical of the slave states? My TLDR is that the majority of the arrests were of Southerners or Border Staters who, in some way, materially supported the Confederacy. The pattern of the arrests is entirely inconsistent with arrests for mere political dissent, especially the type of dissent that a radical abolitionist might perform.
Now for the details: One analysis of arbitrary arrests during this period shows that they were overwhelmingly concentrated in slave states, especially Maryland:
The most notable feature of the arrests is their geographic
distribution. Of the 866 persons on the list, place of residence is
noted for 582. One hundred and seventy-one were Marylanders. Thus a
state with a little over 3 percent of the North's population produced
29.4 percent of the persons arrested by the Lincoln administration in the first year of the war.
A near equal percentage of the arrests were of citizens of Confederate states:
Another substantial percentage of persons arrested were not
Northerners at all. Seventy-nine of the 582, or 13.6 percent were
Virginia residents, and another 12.7 percent were residents of other
seceded states. Thus a whopping 26.3 percent of the people arrested
were citizens of Confederate states, either persons arrested for
causing trouble in the few areas of the South controlled by Union
armies in the first year of the war or Southerners trapped in the
North when the war began and arrested for trying to get back home to
join the Confederacy.
Geography aside, many arrests had nothing to do with political dissent:
Examining the reasons for the arrests gives further cause to
appreciate their lack of impact on Lincoln's political fortunes and
helps provide some measure of their efficiency. Among the 866 arrests
known to have occurred under Seward, 612 have some notation of cause
of the arrest. Many of them have nothing to do with political dissent
— or any other activity which normally takes place on dry land. One
hundred and fourteen (or 18.6 percent) were picked up in boats or
immediately upon disembarking from a vessel.
What were they arrested for? Materially supporting the Confederacy:
It should be remembered too that many of the arrests involved
allegations, not of victimless crimes like holding the wrong political
ideas, but of serious ones like murdering pickets, bushwhacking,
burning bridges, and raising money and men for the Confederate Army.
This was especially the case in Missouri, Virginia, and Kentucky
(always) and Maryland (at times of invasions of the North). The
likelihood, of course, is that the percentage of serious crimes rose
after 1862 as the Union conquered more and more Southern territory,
just as it is likely that the percentage of the civilians arrested who
were Confederate citizens and not possible voters for or against
The linked study does not mention any abolitionists. It's hard to prove a negative, but unless someone can find an example of an abolitionist who was arrested, I think it is very unlikely.