I want to tell my collegue without being direct that I know she has stabbed me in the back?

She has really really pissed me off. You see I’m a bit shy, and if I come into the office late I find it hard to say good morning to people who are already in. I can't raise my voice. So I just smile. But if I come in early and there are a few people in the office, I will say morning to everyone.

My collegue thinks I don't say good morning or goodbye when I leave. She will stalk me and takes in every detail. She is also trying to get rid of me. She always tells me she hates her job and I told her its not a piece of cake for me either. So she says tells the manager you're not happy and quit.

My manager called me in saying she realises that this enviroment might not be for me but I have to make an effort saying good morning and buy as I leave. So this woman has said something about me.

Can I tell this collegue that "I realise someone has grassed on me even though I never done anythingthing to hurt anyone. I don't intend to make friends with everyone I'm just here to do my job."
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Your suspicions are probably correct - that she has told your manager that you don't like your job or something. Why not talk to your manager and tell him/her exactly what happened - how the conversation started. Your colleague said that she dislikes her job, and you responded, "It's not a piece of cake for me either." That will at least put it in perspective in your manager's eyes.

Tell the manager that the job is very important to you, and you want to make an effort. Tell the manager that you fear that your colleague took things out of context and you want to resolve the situation before it gets worse.

And, unfortunately, as painful as it is, if your manager told you to say Good Morning to everyone in the morning, then you'll probably have to. Just say it once, loudly enough that people hear it - you don't have to address everyone individually.

Perhaps you could set up a meeting with both your manager and coworker AFTER you have already talked to the manager. Ask if you should have a joint meeting. Good luck with all this.
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I suggest you meet her over a coffee and let her know your thoughts politely.
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Are you sure the manager herself hasn't noticed you are rather silent?
The best way is to make an effort, perhaps others find it hurtful you don't say hello when you arrive in the morning, or leave at night.
Take it as an opportunity of getting to know all your colleagues a bit better, if you know each one, you won't feel so embarrassed at greeting them as a group when they're all together.
And just ignore the woman, if by any chance she hasn't grassed, you will be making yourself look like a fool, and putting yourself in the wrong.
If she has, she'll feel you know, anyway.
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you really must say good morning to your colleagues. If you don't greet people they assume the worst and will start thinking that you dislike them. This will be the same anywhere you are.

In this little village in France even when I go into a shop I have to greet everyone, whether I know them or not. The shop assistants will say hello to everyone who comes in, the owner comes up to shake hands. Women and small children that we know expect to be kissed on the cheek, male friends shake hands. Just imagine if you worked here, kissing all your colleagues when you arrive and leave the office! The least you can do is say good morning and bye.

Tell your manager that you are a naturally quiet person and that does not mean you don't like it there.
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The best thing to do is to let it go. I work in an office where most people say at least good morning and bye. The few that don't, we all have the Perception that they have no manners. You don't have to make a big production out of it. Just a simple good morning, and a "bye everybody" at night is all you need. It's not to much to ask that everyone just extend the common pleasantries. You wouldn't think twice about saying thank you to someone who helped you. This is a simple request that doesn't take a whole lot of effort. They are not telling you to bring in breakfast for everybody every morning for heavens sake. You can do this, and be better for it.
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She sounds like a nasty bully but perhaps other people do think you're being unfriendly by not saying hello/goodbye.

I know because I have EXACTLY the same problem; I'm not good at projecting my voice and small talk doesn't come naturally.

It took me a bit of practice but I always say "Morning" when I get in and try to say it to anyone else who comes in after me.

And I always make sure I say "See you tomorrow/next week" to at least one person at the end of the day ( I never get up and walk out unless there is someone close by to say goodbye to...)

For the 8 hours in between I just work away and no-one bothers me