When an object in the space comes close to a massive body (like a black hole or a neutron star), we see from Earth that its speed significantly increases while it is close to the body. As we know, this is due to space distortion around massive objects known as gravity. But also, gravity causes time dilation, so an observer near a black hole will experience less time spans between same events than a far-away observer. So, the question is: if we travel on an asteroid and come close to a black hole, will we experience an increase in our speed, or the time dilation will compensate space distortion and we'll feel the same movement speed as before (i. e. no acceleration)?
If you jump out an aeroplane, do you experience high speed? Many people would say you do, since your velocity relative to the ground increases.
Note the importance of the phrase "relative to the ground" since to describe speed you always need to say what you are measuring the velocity relative to.
A simple choice is to measure your velocity relative to a distant observer. In this case then, yes as you pass close to the black hole your velocity relative to the distant observer increases. As you approach the event horizon, you velocity relative to the observer will approach the speed of light.
Depending on how close you pass the the event horizon, you will experience significant time dialation, so that more time will pass in the observer's frame than in yours. This means that if you manage to escape the gravitational pull of the black hole, and return to the observer, you will have found that the observer has aged compared to you.