Generally, yes you can get stronger just by doing bodyweight exercises at home. Kron Gracie is one of the advocates of calisthenics and we all know his results. If you are planning on starting a calisthenics S&C program, there are
First of all let’s have a look at bodyweight exercises. Bodyweight exercises or calisthenics have advantages and disadvantages. Among the advantages we can count:

Being able to execute them anywhere. Push ups, pull ups, air squats, glute bridges, etc require little to no equipment and can be performed pretty much anywhere (perhaps except for the pull ups). As such, if you’re ever short on time and can’t make it to the gym, calisthenics can provide you with a quick workout. This also comes in handy for people who cannot afford both a gym membership along with the Jiu Jitsu membership.

Figure 1. The push up. The starting point for calisthenics.

Compound exercises. Most bodyweight exercises can be considered compound exercises, as they work more than one muscle group, often involving at least 3 or 4 major muscle groups. For example the pull ups involve the upper back and biceps for pulling, shoulders and core for stability and the forearms for the grip.

Very little risk of injury. Another advantage of calisthenics is that they posses very little risk of injury, compared to weightlifting. That does not mean you can safely skip the warmup though.
Easy way of doing circuits. Bodyweight exercises can also easily be combined into circuit-style workouts, to provide you with a full body training session in no time.

Starting point of gymnastics. Calisthenics are a great way to get into gymnastics. Gymnastic skills can provide immense benefits to your Jiu Jitsu game if you decide to go there. Georges Saint Pierre is one of the most conditioned athletes in combat sports and is the advocate of such exercises.

Figure 2. Calisthenics are the starting point for gymnastic skills

Now, let’s also mention some disadvantages of bodyweight training:

Limited loading ability. This poses a big problem for lower body exercises. After you get in a decent shape, air squats will no longer provide you with any strength increases, only increases in muscular endurance. Generally, you can load 10-20kgs in a backpack and make it harder but it’s not always possible and loading for push ups is even harder as the weight on top of you has to be stable. This is one of the main disadvantages, which makes you work more on the muscular endurance continuum once you get in decent shape, rather than strength. A way to circumvent this is use harder variations for the exercises you pick.

Isolation. While they generally work the entire body, if you truly have a weak spot or area that was injured and requires more attention, isolation will be hard to do with calisthenics.

Now that we’ve talked a bit about the pros and cons of calisthenics, let’s get back to the main question. You definitely can get stronger for BJJ with calisthenics however, take into account a few things:

Training state. If you are a newcomer or haven’t exercised much prior to this point, calisthenics will be a great starting point and you will definitely feel increases in strength, endurance, coordination, balance and pretty much all attributes. However if you’re generally well conditioned, calisthenics may only provide you with a plus on the muscular endurance side and you’ll need weight lifting to increase your strength. This is especially valid if we’re talking about the lower body.

Frequency and volume. Any S&C program you run in parallel with grappling sessions has to be carefully programmed so that it doesn’t cause overtraining and/or degrade your grappling training sessions. It’s generally accepted that 2 to 3 conditioning sessions a week is enough for a grappling athlete. In regards to volume, you will need to find the sweet spot between making gains and recovering in time for the next workout. You have to remember Jiu Jitsu will tax your body as well, going too hard on Tuesday with the calisthenics will leave you sore and with a reduced physical potential for the sparring on Wednesday. Start with a low volume (3-4 sets for each movement pattern) and adjust from there.

Rest. In link with the above point, make sure you get enough rest to recover from these sessions. The point of strength and conditioning sessions (regardless of which way you decide to go about it) is to increase your Jiu Jitsu performance. If your Jiu Jitsu starts to suffer because of always being under recovered, then you have to make some adjustments.

Figure 3. Pull ups, a great way to test upper body strength

Now let’s look at some exercises you can actually do in your workout and some progressions.

Push Up. One of the most basic exercises. It strengthens the triceps, shoulders, chest as well as the core. You can adjust the palms to be more to the outside if you want more chest involvement, or closer to yourself if you want to involve the triceps more. Remember dips are also very good to improve your chest, shoulders and triceps.
Let’s have a look at some push ups variations:

Pull ups. Pull ups are especially important for grappling because they improve your pulling muscles. They work the upper back, biceps, forearms, core and even the shoulders to a lesser degree.
Let’s have a look at some pull ups variations:

Squats. Legs are also very important in Jiu Jitsu. Being light on your feet can prove a massive advantage when passing the guard for example. The squat is a basic exercise to improve your leg strength and endurance.

Core movements. Dragon flag, planks, side planks, there are plenty of ways to train the core with calisthenics. Most of them are isometric holds.

So try to determine your level, pick some exercises that are appropriate for you, start with a low volume and build up from there.

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