I don't see any annotation or configuration option for that. You can use hibernate-validator to validate an object before serializing. Since you don't want to add custom annotations you can modify the default serializer by having your objects validated before serialization.
First create a custom serializer that takes another one as constructor argument and uses hibernate validator to validate objects and throw an exception.
class ValidatingSerializer extends JsonSerializer<Object> {
    private final JsonSerializer<Object> defaultSerializer;
    private final Validator validator;
    ValidatingSerializer(final JsonSerializer<Object> defaultSerializer) {
        ValidatorFactory factory = Validation.buildDefaultValidatorFactory();
        this.validator = factory.getValidator();
        this.defaultSerializer = defaultSerializer;
    }
    @Override
    public void serialize(Object value, JsonGenerator gen, SerializerProvider provider)
            throws IOException {
        if (!validator.validate(value).isEmpty()) {
            throw new IOException("Null value encountered");
        }
        defaultSerializer.serialize(value, gen, provider);
    }
}
Then create serializer modifier that will use this serializer:
class ValidatingSerializerModifier extends BeanSerializerModifier {
    @Override
    public JsonSerializer<?> modifySerializer(SerializationConfig config,
             BeanDescription beanDesc, JsonSerializer<?> serializer) {
        return new ValidatingSerializer((JsonSerializer<Object>) serializer);
    }
}
Finally register this on you ObjectMapper with a SimpleModule:
SimpleModule module = new SimpleModule();
module.setSerializerModifier(new ValidatingSerializerModifier());
ObjectMapper mapper = new ObjectMapper();
mapper.registerModule(module);
This will now be used and exceptions will be thrown whenever validation fails for fields annotated with standard validation annotations:
@NotNull // BeanValidation
private String foo;