Query and filter key names instead of values in MongoDB

You can do that using mapReduce:

To get just the field names at root level:

db.collection.mapReduce(function () {
    Object.keys(this).map(function(key) {
        if (key.match(/^fk/)) emit(key, null);

        // OR: key.indexOf("fk") === 0
    });
}, function(/* key, values */) {
    // No need for params or to return anything in the 
    // reduce, just pass an empty function.
}, { out: { inline: 1 }});

This will output something like this:

{
    "results": [{
        "_id": "fkKey1",
        "value": null
    }, {
        "_id": "fkKey2",
        "value": null
    }, {
        "_id": "fkKey3",
        "value": null
    }],
    "timeMillis": W,
    "counts": {
        "input": X,
        "emit": Y,
        "reduce": Z,
        "output": 3
    },
    "ok" : 1
}

To get field names and any or all (whole doc) its values:

db.test.mapReduce(function () {
    var obj = this;

    Object.keys(this).map(function(key) {
        // With `obj[key]` you will get the value of the field as well.
        // You can change `obj[key]` for:
        //  - `obj` to return the whole document.
        //  - `obj._id` (or any other field) to return its value.

        if (key.match(/^fk/)) emit(key, obj[key]);
    });
}, function(key, values) {
    // We can't return values or an array directly yet:

    return { values: values };
}, { out: { inline: 1 }});

This will output something like this:

{
    "results": [{
        "_id": "fkKey1",
        "value": {
            "values": [1, 4, 6]
        }
    }, {
        "_id": "fkKey2",
        "value": {
            "values": ["foo", "bar"]
        }
    }],
    "timeMillis": W,
    "counts": {
        "input": X,
        "emit": Y,
        "reduce": Z,
        "output": 2
    },
    "ok" : 1
}

To get field names in subdocuments (without path):

To do that you will have to use store JavaScript functions on the Server:

db.system.js.save({ _id: "hasChildren", value: function(obj) {
    return typeof obj === "object";
}});

db.system.js.save({ _id: "getFields", value: function(doc) {
    Object.keys(doc).map(function(key) {
        if (key.match(/^fk/)) emit(key, null);

        if (hasChildren(doc[key])) getFields(doc[key])
    });
}});

And change your map to:

function () {
    getFields(this);
}

Now run db.loadServerScripts() to load them.

To get field names in subdocuments (with path):

The previous version will just return field names, not the whole path to get them, which you will need if what you want to do is rename those keys. To get the path:

db.system.js.save({ _id: "getFields", value: function(doc, prefix) {
    Object.keys(doc).map(function(key) {
        if (key.match(/^fk/)) emit(prefix + key, null);

        if (hasChildren(doc[key]))
            getFields(doc[key], prefix + key + '.')
    });
}});

And change your map to:

function () {
    getFields(this, '');
}

To exclude overlapping path matches:

Note that if you have a field fkfoo.fkbar, it will return fkfoo and fkfoo.fkbar. If you don't want overlapping path matches, then:

db.system.js.save({ _id: "getFields", value: function(doc, prefix) {
    Object.keys(doc).map(function(key) {
        if (hasChildren(doc[key]))
            getFields(doc[key], prefix + key + '.')
        else if (key.match(/^fk/)) emit(prefix + key, null);
    });
}});

Going back to your question, renaming those fields:

With this last option, you get all the paths that include keys that start with fk, so you can use $rename for that.

However, $rename doesn't work for those that contain arrays, so for those you could use forEach to do the update. See MongoDB rename database field within array

Performance note:

MapReduce is not particularly fast thought, so you may want to specify { out: "fk_fields"} to output the results into a new collection called fk_fields and query those results later, but that will depend on your use case.

Possible optimisations for specific cases (consistent schema):

Also, note that if you know that the schema of your documents is always the same, then you just need to check one of them to get its fields, so you can do that adding limit: 1 to the options object or just retrieving one document with findOne and reading its fields in the application level.