Q:

Would agriculture cease to exist after the proliferation of replicators?

In a distant future, humans have colonized every corner of the solar system, and we can easily convert any object into anything else - be it food or a tricorder, you name it. Using these replicators we can control standard and quality down to atomic level. so I reckon that the practice of agriculture will fade away and eventually stop. In this future, Earth had becomes almost inhospitable, and terraforming other terrestrial planets and moons is costly and time consuming.

A:

No, it would not cease. For everyone, anyway.
1. Technology doesn't spread everywhere.
If we live on a planet where some indigenous people have never heard of planes, and so fire arrows at them when they fly overhead - then it's completely possible for these machines to not cycle everywhere.
2. Tradition trumps development.
People oppose GMOs - proven to be completely harmless multiple times - because of speculative and unproven reasoning that they are unhealthy (this is ignoring the moral reasoning against genetic modification which may or may not be justified). If a technology that "magically" churns out food is made, surely some farmers - and some people - will not trust it, and the tradition of agriculture will continue to fuel this market.
3. Farming works fine outside Earth.
While terraforming may be expensive, there has to be a way to replenish food in extraterrestrial colonies. You cannot ship food in cheaply, and people need food to eat. At the most fundamental level, this is farming, and it will develop prior to replicators - so there is no real "need" to have them. Open fields - probably not - but farming can still proceed.
4. The economy may not like it.
If farming works, employs people and drives the economy (Food is more valuable when it's harder to get. Sounds like space.) - then it is reasonable to believe settlements will ban replicated food. Just like the "robots are taking our jobs" idea - it's a handy solution, but people pay the price.
5. Replicators aren't free, and that money is better spent elsewhere.
While the majority of people will probably want these devices, not everyone can afford them. I might want an iPhone 7, but I have a 5, and I'm saving money for other things I need more. Similarly, I might want a replicator for food, but I can buy cheap food at the store, and that money could be used on more worthwhile purchases.
6. Replicators obey physics, which makes them expensive.
To make an apple, you need the materials that go into it - or an equal amount of material to convert into those materials - and a ton of electricity to facilitate the conversion and molecular arrangement of these materials. In extraterrestrial colonies, electricity and matter will be valued. Compare this to normal plants and animals (agriculture) which harvest the raw materials and convert them themselves.

A:

It depends.  A couple factors:
energy
All over the solar system: so what is the energy source? Possibly not solar power, far from the sun.
How much energy do you have? Is it pleantiful or scarce?  How does replicating food compare, budget wise, with farming it?
To farm, you’ll need to replicate contained chambers and infrastructure, but that’s reusable. You need to make light to let the plants grow, and then you need to tend and harvest, and then deal with the waste products.
Depending on hiw much power it takes to make the food directly, this might be a savings or it might be grossly inefficient!
And even so, if energy is in abundance, you might indulge the less-efficient option.
consumables
A plot point in Wil McCarthy’s Queendom of Sol universe is that the replicators can’t make other replicators.  So there is a resource cost associated with using them (since they wear out).  If that’s the case, and if said resource is scarce, then it might be important to defer replicator use where possible.
it depends
The real question isn’t “will they or won’t they”, but “under what conditions will they?”  Nailing down the particulars of your setting will cut down the possibilities, because as asked it’s way too open ended.
Note that the answer could vary from place to place within your solar-systemt-spanning civilization.  Maybe on Mars it’s practical to have the one-time (sunk) cost of greenhouses and the plants just grow in sunlight.  In the Kuper belt, getting plants to grow naturally takes excessive energy, and requires elaborate cetrifuge farms to work at all, and is wholly impractical.

