I'm not really sure why you would want to bother making something that by definition is fatty when you are trying to avoid fatty foods.
Egg substitutes simply don't have the same flavour or richness.
So, yes, you could do that.
It just won't taste recognizably like quiche.
You may suffer from denser texture, also.
cooking.stackexchange.com/users/None/
Ener-G and Bob's makes an Egg replacer that is used as a substitute in many dishes.
I have not tried it in quiche specifically
but it works well in other baking applications
(it's not an egg substitute like Egg-Beaters, which would be another substitute).
If your primary aim is to reduce the fat, you might consider just using the whites of the egg and adding more savory ingredients to supplement the loss of the yolk flavoring.
The actual body of the quiche will not suffer (the structure is due to the whites); it might throw off the ratios of how much [egg] yields how much [cake].
Also, you can omit or substitute the cheese for low-fat options, change the crust to a less fattening one (use vegetable shortening as opposed to lard), substitute butter for Earth Balance, etc.
cooking.stackexchange.com/users/15/mfg
Although I understand what Daniel is saying about it not tasting the same as a traditional quiche I disagree in that I think trying to look for alternative ways to have 'similar' recipes to those you need to avoid for health reasons is an excellent idea.
I often cook a 'crustless' pizza in order to cut out the bread and you can do something similar to cut out the pastry of the quiche (as rumtscho said, this is the part that contains the most saturated fat).
The recipes tend to require eggs, but you will be cutting out the pastry and it should have more of a 'quichey' taste.
If you want to go even further and cut out the eggs you can use a vegan quiche recipe (although you will be compromising on the traditional taste).
An example of a crustless quiche that includes eggs is here- http://allrecipes.co.uk/recipe/1920/easy-mushroom-quiche.aspx
cooking.stackexchange.com/users/5162/nixy 
If you're really worried about fat and cholesterol, you shouldn't be eating quiche, period.
You might be able to make something that's low in fat, but it won't be a quiche anymore.
cooking.stackexchange.com/users/5602/Mike Baranczak
Not that I have made any, but I have eaten some - the Spanish make frittatas, which are like crustless quiche! http://www.bbcgoodfood.com/search.do?keywords=frittata
Perhaps you should look into that -
cooking.stackexchange.com/users/5649/Charlotte Farley
I'd like to make a quiche this weekend (for the first time!), but I really want to reduce some of the fat and cholesterol due to health issues. Can I use an egg substitute (one of those that comes in a carton) instead of whole eggs in a quiche?
Egg Substitutions in a Quiche
cooking.stackexchange.com
<substitutions><eggs><quiche>
cooking.stackexchange.com/questions/13855
cooking.stackexchange.com/users/3113/Alison S
While proper quiche is fatty by necessity, you could make an acceptable alternative with an egg replacer by Ener-G and Bob's, omit or substitute the cheese with a low-fat option, use vegetable shortening instead of lard or try something like a crustless frittata for a similar flavour. Egg substitutes don't give good results but you can try Ener-G and Bob's egg replacers. You can also try substituting the cheese or making a crustless quiche for lower cholesterol options.
Egg substitutes will not give you the same result. You should just abandon eating quiche. Ener-G and Bob's are good egg replacers. Try substituting the cheese or making a crustless quiche instead.